Topic 32: Two-Way Mixed
Effects Model
Outline
• Two-way mixed models
• Three-way mixed models
Data for two-way design
• Y is the response variable
• Factor A with levels i = 1 to a
• Factor B with levels j = 1 to b
• Yijk is the kth observation in cell (i, j)
k = 1 to nij
• Have balanced designs with n = nij
Two-way mixed model
• Two-way mixed model has
– One fixed effect
– One random effect
• Tests:
– Again use EMS as guide
– Two possible models
• Unrestricted mixed model (SAS)
• Restricted mixed model (Text)
KNNL Example
• KNNL Problem 25.15, p 1080
• Y is fuel efficiency in miles per gallon
• Factor A represents four different
drivers, a=4 levels
• Factor B represents five different cars
of the same model , b=5
• Each driver drove each car twice over
the same 40-mile test course
Read and check the data
data a1;
infile 'c:\...\CH25PR15.TXT';
input mpg driver car;
proc print data=a1;
run;
The data
Obs
1
2
3
4
5
6
7
8
9
10
mpg
25.3
25.2
28.9
30.0
24.8
25.1
28.4
27.9
27.1
26.6
driver
1
1
1
1
1
1
1
1
1
1
car
1
1
2
2
3
3
4
4
5
5
Prepare the data for a plot
data a1; set a1;
if (driver eq 1)*(car eq 1)
then dc='01_1A';
if (driver eq 1)*(car eq 2)
then dc='02_1B';
⋮
if (driver eq 4)*(car eq 5)
then dc='20_4E';
Plot the data
title1 'Plot of the data';
symbol1 v=circle i=none c=black;
proc gplot data=a1;
plot mpg*dc/frame;
run;
Find the means
proc means data=a1;
output out=a2 mean=avmpg;
var mpg;
by driver car;
Plot the means
title1 'Plot of the means';
symbol1 v='A' i=join c=black;
symbol2 v='B' i=join c=black;
symbol3 v='C' i=join c=black;
symbol4 v='D' i=join c=black;
symbol5 v='E' i=join c=black;
proc gplot data=a2;
plot avmpg*driver=car/frame;
run;
Example Revisited
• Suppose that the four drivers were not
randomly selected and there is interest
in comparing the four drivers in the
study
• Driver (A) is now a fixed effect
• Still consider Car (B) to be a random
effect
Mixed effects model
(unrestricted)
• Yijk = μ + i + j + ()ij + εijk
• Σi =0 (unknown constants)
• j ~ N(0, σ2)
• ()ij ~ N(0, σ2)
• εij ~ N(0, σ2)
• σY2 = σ2 + σ2 + σ2
Mixed effects model
(restricted)
• Yijk = μ + i + j + ()ij + εijk
• Σi =0 (unknown constants)
a 1 2
2
•  j ~ N (0,   ) and ( )ij ~ N (0, a   )
• Σ(b)ij =0 for all j
• εij ~ N(0, σ2)
• σY2 = σ2 + ((a-1)/a)σ2 + σ2
Parameters
• There are a+3 parameters in this
model
– a fixed effects means
– σ 2
– σ2
– σ2
ANOVA table
• The terms and layout of the ANOVA
table are the same as what we used
for the fixed effects model
• The expected mean squares (EMS)
are different and vary based on the
choice of unrestricted or restricted
mixed model
EMS (unrestricted)
•
•
•
•
•
E(MSA) = σ2 + bnΣi2 /(a-1)+ nσ2
E(MSB) = σ2 + anσ2 + nσ2
E(MSAB) = σ2 + nσ2
E(MSE) = σ2
Estimates of the variance components
can be obtained from these equations,
replacing E(MS) with table value, or other
methods such as ML
EMS (restricted)
•
•
•
•
•
E(MSA) = σ2 + bnΣi2 /(a-1)+ nσ2
E(MSB) = σ2 + anσ2
Diff
2
2
E(MSAB) = σ + nσ
here
2
E(MSE) = σ
Estimates of the variance components
can be obtained from these equations,
replacing E(MS) with table value, or other
methods such as ML
Hypotheses (unrestricted)
• H0A: σ2 = 0; H1A: σ2 ≠ 0
– H0A is tested by F = MSA/MSAB with df
a-1 and (a-1)(b-1)
• H0B: σ2 = 0; H1B : σ2 ≠ 0
– H0B is tested by F = MSB/MSAB with df
b-1 and (a-1)(b-1)
• H0AB : σ2 = 0; H1AB : σ2 ≠ 0
– H0AB is tested by F = MSAB/MSE with df
(a-1)(b-1) and ab(n-1)
Hypotheses (restricted)
• H0A: σ2 = 0; H1A: σ2 ≠ 0
– H0A is tested by F = MSA/MSAB with df
a-1 and (a-1)(b-1)
• H0B: σ2 = 0; H1B : σ2 ≠ 0
– H0B is tested by F = MSB/MSE with df
b-1 and ab(n-1)
• H0AB : σ2 = 0; H1AB : σ2 ≠ 0
– H0AB is tested by F = MSAB/MSE with df
(a-1)(b-1) and ab(n-1)
Comparison of Means
• To compare fixed levels of A, std
error is
2MSAB / bn
• Degrees of freedom for t tests and
CIs are then (a-1)(b-1)
• This is true for both unrestricted and
restricted mixed models
Using Proc Mixed
proc mixed data=a1;
class car driver;
model mpg=driver;
random car car*driver / vcorr;
lsmeans driver / adjust=tukey;
run;
SAS considers unrestricted model
only…results in slightly different variance
estimates
SAS Output
Covariance Parameter
Estimates
Cov Parm
Estimate
car
2.9343
car*driver
0.01406
Residual
0.1757
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
driver
3 12 458.26 <.0001
SAS Output
Effect
driver
driver
driver
driver
driver
1
2
3
4
Least Squares Means
Standard
Estimate
Error DF t Value Pr > |t|
26.9300
0.7793 12
34.56 <.0001
34.1500
0.7793 12
43.82 <.0001
28.8500
0.7793 12
37.02 <.0001
30.2600
0.7793 12
38.83 <.0001
SAS Output
Differences of Least Squares Means
Effect driver _driver
driver 1
2
driver 1
3
driver 1
4
driver 2
3
driver 2
4
driver 3
4
Standard
Error DF t Value Pr > |t| Adjustment Adj P
Estiate
<.0001
-7.2200
0.2019 12 -35.76 <.0001 TukeyKramer
<.0001
-1.9200
0.2019 12 -9.51 <.0001 TukeyKramer
<.0001
-3.3300
0.2019 12 -16.49 <.0001 TukeyKramer
<.0001
5.3000
0.2019 12 26.25 <.0001 TukeyKramer
<.0001
3.8900
0.2019 12 19.26 <.0001 TukeyKramer
<.0001
-1.4100
0.2019 12 -6.98 <.0001 TukeyKramer
Three-way models
• We can have zero, one, two, or three
random effects
• EMS indicate how to do tests
• In some cases the situation is
complicated and we need
approximations of an F test, e.g. when
all are random, use MS(AB)+MS(AC)MS(ABC) to test A
Last slide
• Finish reading KNNL Chapter 25
• We used program topic32.sas to
generate the output for today